Theorem cocanfo 35876

Description: Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Assertion

Ref	Expression
cocanfo	⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐺 = 𝐻)

Proof of Theorem cocanfo

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 766	. . . . . 6 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹))
2	1	fveq1d 6776	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑦) = ((𝐻 ∘ 𝐹)‘𝑦))
3		simpl1 1190	. . . . . . 7 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐹:𝐴–onto→𝐵)
4		fof 6688	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
5	3, 4	syl 17	. . . . . 6 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐹:𝐴⟶𝐵)
6		fvco3 6867	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑦) = (𝐺‘(𝐹‘𝑦)))
7	5, 6	sylan 580	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑦) = (𝐺‘(𝐹‘𝑦)))
8		fvco3 6867	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑦) = (𝐻‘(𝐹‘𝑦)))
9	5, 8	sylan 580	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑦) = (𝐻‘(𝐹‘𝑦)))
10	2, 7, 9	3eqtr3d 2786	. . . 4 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)))
11	10	ralrimiva 3103	. . 3 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → ∀𝑦 ∈ 𝐴 (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)))
12		fveq2 6774	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐺‘(𝐹‘𝑦)) = (𝐺‘𝑥))
13		fveq2 6774	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐻‘(𝐹‘𝑦)) = (𝐻‘𝑥))
14	12, 13	eqeq12d 2754	. . . . 5 ⊢ ((𝐹‘𝑦) = 𝑥 → ((𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)) ↔ (𝐺‘𝑥) = (𝐻‘𝑥)))
15	14	cbvfo 7161	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑦 ∈ 𝐴 (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
16	3, 15	syl 17	. . 3 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → (∀𝑦 ∈ 𝐴 (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
17	11, 16	mpbid 231	. 2 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥))
18		eqfnfv 6909	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) → (𝐺 = 𝐻 ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
19	18	3adant1 1129	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) → (𝐺 = 𝐻 ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
20	19	adantr 481	. 2 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → (𝐺 = 𝐻 ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
21	17, 20	mpbird 256	1 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐺 = 𝐻)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ∘ ccom 5593   Fn wfn 6428  ⟶wf 6429  –onto→wfo 6431  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441

This theorem is referenced by: (None)